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arXiv:2505.07549

Exponential Relative Entropy Decay Along Quantum Markov Semigroups

Melchior Wirth

Published May 12, 2025• 0 citations
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Abstract

We establish the equivalence between exponential decay of the relative entropy along a quantum Markov semigroup and the modified logarithmic Sobolev inequality for general von Neumann algebras. We also extend an intertwining criterion for the modified logarithmic Sobolev inequality to GNS-symmetric quantum Markov semigroups on infinite-dimensional von Neumann algebras.

DOI
10.48550/arxiv.2505.07549

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We establish the equivalence between exponential decay of the relative entropy along a quantum Markov semigroup and the modified logarithmic Sobolev inequality for general von Neumann algebras. We also extend an intertwining criterion for the modified logarithmic Sobolev inequality to GNS-symmetric quantum Markov semigroups on infinite-dimensional von Neumann algebras.
We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.
States of open quantum systems often decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that finite-dimensional quantum Markov semigroups with GNS detailed balance universally obey complete modified logarithmic Sobolev inequalities (CMLSIs), yielding exponential decay of relative entropy to a subspace of fixed point states. We analyze continuous processes that combine dissipative with Hamiltonian time-evolution, precluding this notion of detailed balance. First, we find counterexamples to CMLSI-like decay for these processes and determine conditions under which it fails. In contrast, we prove that despite its absence at early times, exponential decay re-appears for unital, finite-dimensional quantum Markov semigroups at finite timescales. Finally, we show that when dissipation is much stronger than Hamiltonian time-evolution, the rate of eventual, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone. Dubbed self-restricting noise, this inverse relationship arises when strong damping suppresses effects that would otherwise spread noise beyond its initial subspace.
We establish a relation between the exponential decay rates of quantum Markov semigroups with respect to different inner products. More precisely, it was conjectured by Fagnola, Poletti, Sasso and Umanità that for a Gaussian quantum Markov semigroup, the exponential decay rate with respect to the KMS inner product is bounded below by the exponential decay rate for the GNS inner product. We show that this is indeed the case and not limited to Gaussian quantum Markov semigroups, but holds for quantum Markov semigroups with a faithful normal invariant state on arbitrary von Neumann algebras. Additionally, the KMS inner product can be replaced by a whole class of inner products induced by operator monotone functions.
If a set of von Neumann subalgebras has a trivial intersection in finite dimension, then the sum of relative entropies of a given density to its projection in each such algebra is larger than a multiple of its relative entropy to its projection in the trivial intersection. This results in a subadditivity of relative entropy with a dimension and algebra-dependent, multiplicative constant. As a primary application, this inequality lets us derive relative entropy decay estimates in the form of modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents.
For finite-dimensional quantum channels compatible with a faithful conditional expectation, we show that conditional hyperboundedness and strict GNS contraction together yield a relative entropy contraction. For KMS-symmetric quantum Markov semigroups, a conditional 2→q2\to q2→q hyperboundedness at time t0>0t_0>0t0​>0 combined with a GNS gap λ>0λ>0λ>0 implies a modified logarithmic Sobolev constant α=λ(1−2/q)12(λt0+log⁡M+1−2/q), α=\frac{λ(1-2/q)} {12(λt_0+\log M+1-2/q)},α=12(λt0​+logM+1−2/q)λ(1−2/q)​, where MMM is the hyperboundedness constant. As a consequence, completely bounded hyperboundedness over auxiliary systems gives complete versions of entropy contraction with the same rates.
Abstract We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L2L_2L2​ L 2 -estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.